Related papers: Ehresmann theory and partition monoids
Left Ehresmann monoids, and their two-sided counterpart of Ehresmann monoids, were so named by Lawson, who elucidated their connection to the work of Ehresmann in differential geometry. This article is dedicated to building a theory for…
Right (and left) coherency and right (and left) weak coherency are natural finitary conditions for monoids. Determining whether or not a given monoid has any of these properties is historically a difficult problem. This paper has several…
In this thesis we study the subsemigroup structure of the symmetric inverse monoid $I_X$, the inverse semigroup of bijections between subsets of the set $X$, when $X$ is an infinite set. We explore three different approaches to this task.…
The full-domain partition monoid $P_n^{fd}$ has been discovered independently in two recent studies on connections between diagram monoids and category theory. It is a right restriction Ehresmann monoid, and contains both the full…
A semiring generalises the notion of a ring, replacing the additive abelian group structure with that of a commutative monoid. In this paper, we study a notion positioned between a ring and a semiring -- a semiring whose additive monoid is…
We construct the \emph{inverse partition semigroup} $\mathcal{IP}_X$, isomorphic to the \emph{dual symmetric inverse monoid} $\mathcal{I}^{\ast}_X$, introduced in [6]. We give a convenient geometric illustration for elements of…
We propose a notion of a proper Ehresmann semigroup based on a three-coordinate description of its generating elements governed by certain labelled directed graphs with additional structure. The generating elements are determined by their…
We formulate an alternative approach to describing Ehresmann semigroups by means of left and right \'etale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a…
Let $\mathcal P_X$ and $\mathcal S_X$ be the partition monoid and symmetric group on an infinite set $X$. We show that $\mathcal P_X$ may be generated by $\mathcal S_X$ together with two (but no fewer) additional partitions, and we classify…
This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection…
The primary contribution of this thesis is to introduce and examine the planar modular partition monoid for parameters $m, k \in \mathbb{Z}_{>0}$, which has simultaneously and independently generated interest from other researchers as…
We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let $S$ be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category…
Given a monoid $S$ with $E$ any non-empty subset of its idempotents, we present a novel one-sided version of idempotent completion we call left $E$-completion. In general, the construction yields a one-sided variant of a small category…
We give a complete description of the congruences on the partition monoid $P_X$ and the partial Brauer monoid $PB_X$, where $X$ is an arbitrary infinite set, and also of the lattices formed by all such congruences. Our results complement…
We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point…
This work proposes an alternative approach to the so-called lattice of embedded subsets, which is included in the product of the subset and partition lattices of a finite set, and whose elements are pairs consisting of a subset and a…
The authors [3] proved that the endomorphism semiring of a nontrivial semilattice is always subdirectly irreducible and described its monolith. Here we prove that the endomorphism semiring of a commutative inverse semigroup with at least…
We study monoids generated by various combinations of idempotents and one- or two-sided units of an infinite partial Brauer monoid. This yields a total of eight such monoids, each with a natural characterisation in terms of relationships…
We introduce the inverse monoid of inner partial automorphisms of a semigroup -- a tool that associates to every semigroup an inverse semigroup. When the semigroup is a group, this inverse semigroup is isomorphic to the group of inner…
Mitsch's natural partial order on the semigroup of binary relations is here characterised by equations in the theory of relation algebras. The natural partial order has a complex relationship with the compatible partial order of inclusion,…